847/845: Difference between revisions
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'''847/845''', the '''cuthbert comma''', is a [[13-limit]] (also [[5.7.11.13 subgroup]]) [[comma]] measuring about 4.09{{cent}}. It is the difference between [[7/5]] and a stack of two [[13/11]]'s. It is also the difference between [[125/121]] and [[175/169]] | '''847/845''', the '''cuthbert comma''', is a [[small comma|small]] [[13-limit]] (also [[5.7.11.13 subgroup|5.7.11.13-subgroup]]) [[comma]] measuring about 4.09{{cent}}. It is the difference between [[7/5]] and a stack of two [[13/11]]'s. It is also the difference between [[125/121]] and [[175/169]]. | ||
In terms of full 13-limit commas, it is the difference between the following superparticular pairs: | In terms of full 13-limit commas, it is the difference between the following superparticular pairs: | ||
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* [[385/384]] and [[4225/4224]] | * [[385/384]] and [[4225/4224]] | ||
Meanwhile, it can be factorized as [[1001/1000]] | Meanwhile, it can be factorized as ([[1001/1000]])⋅([[2200/2197]]) or ([[441/440]])⋅([[10648/10647]]). | ||
In the 5.7.11.13 subgroup, it is the simplest comma of its size (and the smallest of its complexity) by an extremely large margin. For comparison, [[637/625]] is about as simple but much larger, and [[2941225/2924207]] is significantly more complex yet still twice as large. | |||
== Temperaments == | == Temperaments == | ||
[[Tempering out]] this comma in the 13-limit results in the rank-5 cuthbert temperament and enables the [[cuthbert chords]]. | [[Tempering out]] this comma in the 13-limit results in the rank-5 '''cuthbert''' temperament and enables the [[cuthbert chords]]. | ||
Tempering it out in the 5.7.11.13 subgroup leads to an extremely efficient rank-3 temperament. | Tempering it out in the 5.7.11.13 subgroup leads to an extremely efficient rank-3 temperament. | ||
[[Category:Cuthbert]] | [[Category:Cuthbert]] | ||
[[Category:Commas with unknown etymology]] | [[Category:Commas with unknown etymology]] | ||
Revision as of 09:02, 9 April 2026
| Interval information |
847/845, the cuthbert comma, is a small 13-limit (also 5.7.11.13-subgroup) comma measuring about 4.09 ¢. It is the difference between 7/5 and a stack of two 13/11's. It is also the difference between 125/121 and 175/169.
In terms of full 13-limit commas, it is the difference between the following superparticular pairs:
Meanwhile, it can be factorized as (1001/1000)⋅(2200/2197) or (441/440)⋅(10648/10647).
In the 5.7.11.13 subgroup, it is the simplest comma of its size (and the smallest of its complexity) by an extremely large margin. For comparison, 637/625 is about as simple but much larger, and 2941225/2924207 is significantly more complex yet still twice as large.
Temperaments
Tempering out this comma in the 13-limit results in the rank-5 cuthbert temperament and enables the cuthbert chords.
Tempering it out in the 5.7.11.13 subgroup leads to an extremely efficient rank-3 temperament.